Since \((G,+,\circ )\)  has cyclic additive group and abelian multiplicative group, by [1]} there exist \(d\in \mathbb {N}\setminus \lbrace 1\rbrace \)  such that \(d=|Soc(G)|\)  and \(g_1\circ g_2=g_1+g_2+g_1 g_2 d\)  for all \(g_1,g_2\in G\) . By [2]}, we have that \(d\)  is a nilpotent element of the ring \((G,+,*)\) . Now, since \(\lambda _{g_1}(g)=(1+dg_1)g\)  for all \(g,g_1\in G\)  and by [1]} for all \(e\in E\)  there exist \(k_e\in \mathbb {N}\)  such that \(\alpha _e(g)=(1+k_e s) g\)  for all \(g\in G\) , by nilel, we have that
\((g_1,e_1)(g_2,e_2)=([(1+dg_1)(1+k_{e_1}s)-1]g_2,0)=((k_{e_1} s+dg_1+dk_{e_1} g_1 s)g_2,0)\) 
